P. 64 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6301-6400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	enpr1g 6301	{𝐴, 𝐴} has only one element. (Contributed by FL, 15-Feb-2010.)
⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐴} ≈ 1𝑜)
Theorem	en1 6302*	A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.)
⊢ (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
Theorem	en1bg 6303	A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Jim Kingdon, 13-Apr-2020.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ≈ 1𝑜 ↔ 𝐴 = {∪ 𝐴}))
Theorem	reuen1 6304*	Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} ≈ 1𝑜)
Theorem	euen1 6305	Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (∃!𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≈ 1𝑜)
Theorem	euen1b 6306*	Two ways to express "𝐴 has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
⊢ (𝐴 ≈ 1𝑜 ↔ ∃!𝑥 𝑥 ∈ 𝐴)
Theorem	en1uniel 6307	A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
⊢ (𝑆 ≈ 1𝑜 → ∪ 𝑆 ∈ 𝑆)
Theorem	2dom 6308*	A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.)
⊢ (2𝑜 ≼ 𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)
Theorem	fundmen 6309	A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ 𝐹 ∈ V    ⇒   ⊢ (Fun 𝐹 → dom 𝐹 ≈ 𝐹)
Theorem	fundmeng 6310	A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.)
⊢ ((𝐹 ∈ 𝑉 ∧ Fun 𝐹) → dom 𝐹 ≈ 𝐹)
Theorem	cnven 6311	A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 ≈ ◡𝐴)
Theorem	fndmeng 6312	A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐴 ≈ 𝐹)
Theorem	en2sn 6313	Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵})
Theorem	snfig 6314	A singleton is finite. (Contributed by Jim Kingdon, 13-Apr-2020.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Fin)
Theorem	fiprc 6315	The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.)
⊢ Fin ∉ V
Theorem	unen 6316	Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))
Theorem	enm 6317*	A set equinumerous to an inhabited set is inhabited. (Contributed by Jim Kingdon, 19-May-2020.)
⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑦 𝑦 ∈ 𝐵)
Theorem	xpsnen 6318	A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 × {𝐵}) ≈ 𝐴
Theorem	xpsneng 6319	A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × {𝐵}) ≈ 𝐴)
Theorem	xp1en 6320	One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 × 1𝑜) ≈ 𝐴)
Theorem	endisj 6321*	Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅)
Theorem	xpcomf1o 6322*	The canonical bijection from (𝐴 × 𝐵) to (𝐵 × 𝐴). (Contributed by Mario Carneiro, 23-Apr-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥})    ⇒   ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
Theorem	xpcomco 6323*	Composition with the bijection of xpcomf1o 6322 swaps the arguments to a mapping. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ 𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥})    &   ⊢ 𝐺 = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐴 ↦ 𝐶)    ⇒   ⊢ (𝐺 ∘ 𝐹) = (𝑧 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
Theorem	xpcomen 6324	Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)
Theorem	xpcomeng 6325	Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
Theorem	xpsnen2g 6326	A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × 𝐵) ≈ 𝐵)
Theorem	xpassen 6327	Associative law for equinumerosity of Cartesian product. Proposition 4.22(e) of [Mendelson] p. 254. (Contributed by NM, 22-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ ((𝐴 × 𝐵) × 𝐶) ≈ (𝐴 × (𝐵 × 𝐶))
Theorem	xpdom2 6328	Dominance law for Cartesian product. Proposition 10.33(2) of [TakeutiZaring] p. 92. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 ≼ 𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
Theorem	xpdom2g 6329	Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐴 ≼ 𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
Theorem	xpdom1g 6330	Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐴 ≼ 𝐵) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶))
Theorem	xpdom3m 6331*	A set is dominated by its Cartesian product with an inhabited set. Exercise 6 of [Suppes] p. 98. (Contributed by Jim Kingdon, 15-Apr-2020.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ ∃𝑥 𝑥 ∈ 𝐵) → 𝐴 ≼ (𝐴 × 𝐵))
Theorem	xpdom1 6332	Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by NM, 29-Mar-2006.) (Revised by Mario Carneiro, 7-May-2015.)
⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 ≼ 𝐵 → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶))
Theorem	fopwdom 6333	Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴)
Theorem	enen1 6334	Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ≈ 𝐶 ↔ 𝐵 ≈ 𝐶))
Theorem	enen2 6335	Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
⊢ (𝐴 ≈ 𝐵 → (𝐶 ≈ 𝐴 ↔ 𝐶 ≈ 𝐵))
Theorem	domen1 6336	Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼ 𝐶 ↔ 𝐵 ≼ 𝐶))
Theorem	domen2 6337	Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
⊢ (𝐴 ≈ 𝐵 → (𝐶 ≼ 𝐴 ↔ 𝐶 ≼ 𝐵))
2.6.26  Pigeonhole Principle
Theorem	phplem1 6338	Lemma for Pigeonhole Principle. If we join a natural number to itself minus an element, we end up with its successor minus the same element. (Contributed by NM, 25-May-1998.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (suc 𝐴 ∖ {𝐵}))
Theorem	phplem2 6339	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
Theorem	phplem3 6340	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. For a version without the redundant hypotheses, see phplem3g 6342. (Contributed by NM, 26-May-1998.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
Theorem	phplem4 6341	Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))
Theorem	phplem3g 6342	A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6340 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
Theorem	nneneq 6343	Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136. (Contributed by NM, 28-May-1998.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵))
Theorem	php5 6344	A natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90. (Contributed by NM, 26-Jul-2004.)
⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ suc 𝐴)
Theorem	snnen2og 6345	A singleton {𝐴} is never equinumerous with the ordinal number 2. If 𝐴 is a proper class, see snnen2oprc 6346. (Contributed by Jim Kingdon, 1-Sep-2021.)
⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} ≈ 2𝑜)
Theorem	snnen2oprc 6346	A singleton {𝐴} is never equinumerous with the ordinal number 2. If 𝐴 is a set, see snnen2og 6345. (Contributed by Jim Kingdon, 1-Sep-2021.)
⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ≈ 2𝑜)
Theorem	1nen2 6347	One and two are not equinumerous. (Contributed by Jim Kingdon, 25-Jan-2022.)
⊢ ¬ 1𝑜 ≈ 2𝑜
Theorem	phplem4dom 6348	Dominance of successors implies dominance of the original natural numbers. (Contributed by Jim Kingdon, 1-Sep-2021.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≼ suc 𝐵 → 𝐴 ≼ 𝐵))
Theorem	php5dom 6349	A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.)
⊢ (𝐴 ∈ ω → ¬ suc 𝐴 ≼ 𝐴)
Theorem	nndomo 6350	Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146. (Contributed by NM, 17-Jun-1998.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵))
Theorem	phpm 6351*	Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. By "proper subset" here we mean that there is an element which is in the natural number and not in the subset, or in symbols ∃𝑥𝑥 ∈ (𝐴 ∖ 𝐵) (which is stronger than not being equal in the absence of excluded middle). Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of lemmas phplem1 6338 through phplem4 6341, nneneq 6343, and this final piece of the proof. (Contributed by NM, 29-May-1998.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵)) → ¬ 𝐴 ≈ 𝐵)
Theorem	phpelm 6352	Pigeonhole Principle. A natural number is not equinumerous to an element of itself. (Contributed by Jim Kingdon, 6-Sep-2021.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ¬ 𝐴 ≈ 𝐵)
Theorem	phplem4on 6353	Equinumerosity of successors of an ordinal and a natural number implies equinumerosity of the originals. (Contributed by Jim Kingdon, 5-Sep-2021.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))
2.6.27  Finite sets
Theorem	fidceq 6354	Equality of members of a finite set is decidable. This may be counterintuitive: cannot any two sets be elements of a finite set? Well, to show, for example, that {𝐵, 𝐶} is finite would require showing it is equinumerous to 1𝑜 or to 2𝑜 but to show that you'd need to know 𝐵 = 𝐶 or ¬ 𝐵 = 𝐶, respectively. (Contributed by Jim Kingdon, 5-Sep-2021.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → DECID 𝐵 = 𝐶)
Theorem	fidifsnen 6355	All decrements of a finite set are equinumerous. (Contributed by Jim Kingdon, 9-Sep-2021.)
⊢ ((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
Theorem	fidifsnid 6356	If we remove a single element from a finite set then put it back in, we end up with the original finite set. This strengthens difsnss 3531 from subset to equality when the set is finite. (Contributed by Jim Kingdon, 9-Sep-2021.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
Theorem	nnfi 6357	Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin)
Theorem	enfi 6358	Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
Theorem	enfii 6359	A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.)
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
Theorem	ssfilem 6360*	Lemma for ssfiexmid 6361. (Contributed by Jim Kingdon, 3-Feb-2022.)
⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	ssfiexmid 6361*	If any subset of a finite set is finite, excluded middle follows. One direction of Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 19-May-2020.)
⊢ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	infiexmid 6362*	If the intersection of any finite set and any other set is finite, excluded middle follows. (Contributed by Jim Kingdon, 5-Feb-2022.)
⊢ (𝑥 ∈ Fin → (𝑥 ∩ 𝑦) ∈ Fin)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	domfiexmid 6363*	If any set dominated by a finite set is finite, excluded middle follows. (Contributed by Jim Kingdon, 3-Feb-2022.)
⊢ ((𝑥 ∈ Fin ∧ 𝑦 ≼ 𝑥) → 𝑦 ∈ Fin)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	dif1en 6364	If a set 𝐴 is equinumerous to the successor of a natural number 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.)
⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
Theorem	fiunsnnn 6365	Adding one element to a finite set which is equinumerous to a natural number. (Contributed by Jim Kingdon, 13-Sep-2021.)
⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ (V ∖ 𝐴)) ∧ (𝑁 ∈ ω ∧ 𝐴 ≈ 𝑁)) → (𝐴 ∪ {𝐵}) ≈ suc 𝑁)
Theorem	php5fin 6366	A finite set is not equinumerous to a set which adds one element. (Contributed by Jim Kingdon, 13-Sep-2021.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ (V ∖ 𝐴)) → ¬ 𝐴 ≈ (𝐴 ∪ {𝐵}))
Theorem	fisbth 6367	Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.)
⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → 𝐴 ≈ 𝐵)
Theorem	0fin 6368	The empty set is finite. (Contributed by FL, 14-Jul-2008.)
⊢ ∅ ∈ Fin
Theorem	fin0 6369*	A nonempty finite set has at least one element. (Contributed by Jim Kingdon, 10-Sep-2021.)
⊢ (𝐴 ∈ Fin → (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴))
Theorem	fin0or 6370*	A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.)
⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ∃𝑥 𝑥 ∈ 𝐴))
Theorem	diffitest 6371*	If subtracting any set from a finite set gives a finite set, any proposition of the form ¬ 𝜑 is decidable. This is not a proof of full excluded middle, but it is close enough to show we won't be able to prove 𝐴 ∈ Fin → (𝐴 ∖ 𝐵) ∈ Fin. (Contributed by Jim Kingdon, 8-Sep-2021.)
⊢ ∀𝑎 ∈ Fin ∀𝑏(𝑎 ∖ 𝑏) ∈ Fin    ⇒   ⊢ (¬ 𝜑 ∨ ¬ ¬ 𝜑)
Theorem	findcard 6372*	Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ Fin → (∀𝑧 ∈ 𝑦 𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ Fin → 𝜏)
Theorem	findcard2 6373*	Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 8-Jul-2010.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ Fin → (𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ Fin → 𝜏)
Theorem	findcard2s 6374*	Variation of findcard2 6373 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ Fin → 𝜏)
Theorem	findcard2d 6375*	Deduction version of findcard2 6373. If you also need 𝑦 ∈ Fin (which doesn't come for free due to ssfiexmid 6361), use findcard2sd 6376 instead. (Contributed by SO, 16-Jul-2018.)
⊢ (𝑥 = ∅ → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓 ↔ 𝜏))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂))    &   ⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝜃 → 𝜏))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    ⇒   ⊢ (𝜑 → 𝜂)
Theorem	findcard2sd 6376*	Deduction form of finite set induction . (Contributed by Jim Kingdon, 14-Sep-2021.)
⊢ (𝑥 = ∅ → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓 ↔ 𝜏))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂))    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝜃 → 𝜏))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    ⇒   ⊢ (𝜑 → 𝜂)
Theorem	diffisn 6377	Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ∈ Fin)
Theorem	diffifi 6378	Subtracting one finite set from another produces a finite set. (Contributed by Jim Kingdon, 8-Sep-2021.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∖ 𝐵) ∈ Fin)
Theorem	ac6sfi 6379*	Existence of a choice function for finite sets. (Contributed by Jeff Hankins, 26-Jun-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	en2eqpr 6380	Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
⊢ ((𝐶 ≈ 2𝑜 ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 ≠ 𝐵 → 𝐶 = {𝐴, 𝐵}))
Theorem	fientri3 6381	Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴))
Theorem	nnwetri 6382*	A natural number is well-ordered by E. More specifically, this order both satisfies We and is trichotomous. (Contributed by Jim Kingdon, 25-Sep-2021.)
⊢ (𝐴 ∈ ω → ( E We 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)))
Theorem	onunsnss 6383	Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.)
⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵 ⊆ 𝐴)
Theorem	unsnfi 6384	Adding a singleton to a finite set yields a finite set. (Contributed by Jim Kingdon, 3-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ ¬ 𝐵 ∈ 𝐴) → (𝐴 ∪ {𝐵}) ∈ Fin)
Theorem	unsnfidcex 6385	The 𝐵 ∈ 𝑉 condition in unsnfi 6384. This is intended to show that unsnfi 6384 without that condition would not be provable but it probably would need to be strengthened (for example, to imply included middle) to fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ 𝐴 ∧ (𝐴 ∪ {𝐵}) ∈ Fin) → DECID ¬ 𝐵 ∈ V)
Theorem	unsnfidcel 6386	The ¬ 𝐵 ∈ 𝐴 condition in unsnfi 6384. This is intended to show that unsnfi 6384 without that condition would not be provable but it probably would need to be strengthened (for example, to imply included middle) to fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ Fin) → DECID ¬ 𝐵 ∈ 𝐴)
Theorem	snon0 6387	An ordinal which is a singleton is {∅}. (Contributed by Jim Kingdon, 19-Oct-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅)
Theorem	fnfi 6388	A version of fnex 5404 for finite sets that does not require Replacement. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)
Theorem	fundmfi 6389	The domain of a finite function is finite. (Contributed by Jim Kingdon, 5-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ Fun 𝐴) → dom 𝐴 ∈ Fin)
Theorem	fundmfibi 6390	A function is finite if and only if its domain is finite. (Contributed by AV, 10-Jan-2020.)
⊢ (Fun 𝐹 → (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin))
Theorem	relcnvfi 6391	If a relation is finite, its converse is as well. (Contributed by Jim Kingdon, 5-Feb-2022.)
⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡𝐴 ∈ Fin)
Theorem	funrnfi 6392	The range of a finite relation is finite if its converse is a function. (Contributed by Jim Kingdon, 5-Feb-2022.)
⊢ ((Rel 𝐴 ∧ Fun ◡𝐴 ∧ 𝐴 ∈ Fin) → ran 𝐴 ∈ Fin)
Theorem	f1dmvrnfibi 6393	A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6394. (Contributed by AV, 10-Jan-2020.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))
Theorem	f1vrnfibi 6394	A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 6393. (Contributed by AV, 10-Jan-2020.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))
2.6.28  Supremum and infimum
Syntax	csup 6395	Extend class notation to include supremum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers.
class sup(𝐴, 𝐵, 𝑅)
Syntax	cinf 6396	Extend class notation to include infimum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers.
class inf(𝐴, 𝐵, 𝑅)
Definition	df-sup 6397*	Define the supremum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the supremum exists. (Contributed by NM, 22-May-1999.)
⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))}
Definition	df-inf 6398	Define the infimum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the infimum exists. For example, 𝑅 could be 'less than', 𝐵 could be the set of real numbers, and 𝐴 could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.)
⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)
Theorem	supeq1 6399	Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
Theorem	supeq1d 6400	Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))